Saving multiple animations in the same action
You can have multiple properties animated in one action. An action will animate the same property on any object it is assigned to, this means you will need one action for each object unless you want them to move exactly the same.

The easiest solution would be to use an armature. You can parent each existing object to a single bone or you can join the objects into one, the mesh pieces don't have to be connected, and have the armature bones move different parts of the mesh. This will bring all of the animation into the one action.